Tautologies & Contradictions

Logical operations and their truth tables

Logical symbols are used to link basic assertions to construct a compound statement. This process is known as logical operations. AND, OR, NOT, Conditional, and Bi-conditional are the five primary logical operations based on respective symbols. Let us study all of the symbols, their meanings, and how they work one by one using truth tables.

• AND Operation
  AND is represented by the symbol “Λ.” A conjunction of two statements occurs when two statements are utilized to produce a compound statement using the AND sign.     
  
  Truth Table:   

                                                                              The truth table for AND Operation

• OR Operation
  OR is represented by the symbol “⋁.”A disjunction of two assertions occurs when two simple statements are utilized to produce a compound statement using an OR sign.     
  
  Truth Table:    

                                                                               The truth table for OR Operation

• Conditional Operation
  The symbol represents conditional Operation “→.”Conditional Operation occurs when a compound statement is generated by two basic assertions linked by the phrase 'if and then.'
  
  Truth Table:    

                                                                        The truth table for Conditional Operation

• Bi-conditional Operation
  Bi-conditional Operation is represented by the symbol “⇔”. Bi-conditional operation occurs when a compound statement is generated by two basic assertions linked by the phrase ‘if and only if’.   
  
  Truth Table:  

                                                                  The truth table for Bi-conditional Operation

• NOT Operation
  A statement's negation occurs when the truth value of a statement is modified using the term NOT. If we consider x a given statement, then ~x is provided by NOT Operation.     
  
  Truth Table:          

                                                                             The truth table for NOT Operation

Tautology

A tautology is a compound assertion that always yields the Truth value in mathematics. The outcome in tautology is always true, regardless of the constituent parts. Contradiction or fallacy is the inverse of tautology, as we will see below. With logical symbols, tautologies may be easily translated from common language to mathematical equations. 

Let us consider the following:

P = I am a coder.

~P = I am not a coder (Since it is the opposite statement of P)

These two separate propositions are linked together using the logical operator "OR," commonly represented by the symbol "⋁ ".

As a result, the above sentence may be expressed as P ⋁ ~P.

Now we'll see if the above statement yields a correct response.

Case 1: I am a coder. In this example, the first statement is correct, whereas the second is incorrect. Because the above statement is connected using the OR operator, it yields TRUE.

Case 2: I am not a coder. The first statement is incorrect, but the second statement is correct. As a result, it yields TRUE.

Now, let us see this case with the help of the truth table.

Thus the final column of the truth table has a TRUE value for every case; hence the given statement can be called a Tautology.

Contradiction

The compound Statement that results in FALSE for every value of their components is known as Contradiction.

Let us consider the following:

P = I have learned DSA.

~P = I have not learned DSA (Since it is the opposite statement of P)

These two separate propositions are linked together using the logical operator "AND" commonly represented by the symbol "Λ".

As a result, the above sentence may be expressed as P Λ ~P.

Now we'll see if the above statement yields a correct response.

Case 1: I have learned DSA. In this example, the first statement is correct, whereas the second is incorrect. Because the above statement is connected using the AND operator, it yields FALSE.

Case 2: I have not learned DSA. The first statement is incorrect, but the second statement is correct. As a result, it yields TRUE.

Now, let us see this case with the help of the truth table.

Thus the final column of the truth table has a FALSE value for every case; hence the given statement can be called a Contradiction.

Contingency

A statement that is neither a tautology nor a contradiction is called a contingency.

As we can see, every value of the last column has both TRUE and FALSE; therefore, it is a contingency. 

FAQs

1. What is a mathematical statement?
   It is the fundamental unit of all mathematical reasoning. Furthermore, mathematical reasoning might be inductive (mathematical induction) or deductive (mathematical deduction). Any forceful language that may be claimed to be true or untrue but not both indicates that it is a mathematically admissible assertion. As a result, this form of statement is correct.
    
2. What is a valid statement?
   A compound proposition is valid when it is a tautology.
    
3. What is satisfiability?
   A compound proposition is satisfiable if at least one TRUE result is in the truth table.

Key Takeaways

This article briefly discussed Tautologies and Contradictions, and we have also discussed some basic terminologies related to them, like logic symbols and truth tables.

I hope you must have gained some insight into this topic of Tautologies and Contradictions, and by now, you must have developed a clear understanding of them. You can learn more about such topics on our platform Coding Ninjas Studio.

Thank you for reading.